Weak coupling limit of the Anisotropic KPZ equation

Abstract

We study the 2-dimensional anisotropic KPZ equation (AKPZ), which is formally given by

$${\partial }_{t}h=\frac{1}{2}\mathrm{\Delta }h+\mathit{\lambda }({({\partial }_{1}h)}^2-{({\partial }_{2}h)}^2)+\mathrm{\xi },$$

where ξ denotes a space-time white noise and $\mathit{\lambda }>0$ is the so-called coupling constant. The AKPZ equation is a critical SPDE, meaning that not only is it analytically ill posed but also the breakthrough pathwise techniques for singular SPDEs in earlier works by Hairer, Gubinelli, Imkeller, and Perkowski are not applicable. As shown in recent work by the authors, the equation regularized at scale N has a diffusion coefficient that diverges logarithmically as the regularization is removed in the limit $N\to \mathrm{\infty }$. Here, we study the weak coupling limit where $\mathit{\lambda }={\mathit{\lambda }}_N=\hat{\mathit{\lambda }}∕\sqrt{logN}$: this is the correct scaling that guarantees that the nonlinearity has a still nontrivial but nondivergent effect. In fact, as $N\to \mathrm{\infty }$ the sequence of equations converges to the linear stochastic heat equation

$${\partial }_{t}h=\frac{{\mathrm{\nu }}_{\mathrm{eff}}}{2}\mathrm{\Delta }h+\sqrt{{\mathrm{\nu }}_{\mathrm{eff}}}\mathrm{\xi },$$

where ${\mathrm{\nu }}_{\mathrm{eff}}>1$ is explicit and depends nontrivially on $\hat{\mathit{\lambda }}$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearized via Cole–Hopf or any other transformation.